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II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it,...
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II. Electromagnetic Waves II. Electromagnetic Waves A. Displacement Current 1. Recall Ampere’s Law: 2. As we’ve learned it, AL is incomplete. We need to add an additional current, called the displacement current, I D . ID arises from time-varying electric fields (not present in a steady current along an infinite wire): enc I L d B 0 dt d I dt E d I E D D ; | | | | (II.A.1,2)
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Transcript of II. Electromagnetic Waves A.Displacement Current 1.Recall Ampere’s Law: 2.As we’ve learned it,...
II. Electromagnetic WavesII. Electromagnetic Waves
A. Displacement Current1. Recall Ampere’s Law:
2. As we’ve learned it, AL is incomplete. We need to add an additional current, called the displacement current, ID. ID arises from time-varying electric fields (not present in a steady current along an infinite wire):
encILdB 0
dt
dI
dt
EdI
ED
D
;||
||(II.A.1,2)
II. Electromagnetic WavesII. Electromagnetic Waves
A. Displacement Current3. General form of Ampere’s Law includes terms
due to “conduction current” and “displacement current”:
encE
c dt
dILdB )( 00
(II.A.3)
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS1. Unified description of E, B:
.)(
.
.0
.
00
0
encE
c
B
enc
dt
dILdB
dt
dLdE
AdB
QAdE
(II.B.1-4)
(Gauss’s Law)
(Gauss’s Law for B)
(Faraday’s Law for B)
(Ampere’s Law for B)
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
V
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
E
B
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS2. Plane Wave
a) As we shall see, the solution to Maxwell’s Equations is a wave of Electric and Magnetic Fields.
b) Plane Wave Definition: Wave in which the transverse components are uniform on a plane perpendicular to the direction of propagation.
E
B
E
B
E
B
II. Electromagnetic WavesII. Electromagnetic Waves
B.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS3. Electromagnetic Wave Properties
a) Transverse waveb) Ratio between E,B:
E/B = c. (II.B.5)
c) Constant speedd) No medium required: E
and B reinforce each other.
E
B
E
B
E
B
E
B
II. Electromagnetic WavesII. Electromagnetic WavesB.B. MAXWELL’S EQUATIONSMAXWELL’S EQUATIONS
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Wavea) Consider a plane wave with Bz, Ey propagating
in the x-direction with speed v. After time t, the two wave fronts are separated by a distance x.
b) Apply Faraday’s Law to a rectangle
in the xy-plane:E
B
E
B
x x x
y
z B, A
a
x
y
)).,(),((
),(),(
txEtxxEa
atxxEatxE
LdE
yy
yy
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Faraday’s Law to a rectangle
in the xy-plane: Assume x is small enough that Bz ~ uniform
over surface.xa
t
txB
dt
d zB
),(
E
B
E
B
x x x
y
z
B, A
a
x
y
x
.),(),(),(
)),(),((t
txB
x
txExa
t
txBtxEtxxEa zyz
yy
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Ampere’s Law to a rectangle
in the zx-plane: Assume x is small enough that Ey ~ uniform
over surface.
E
B
E
B
x x x
y
z
E, A
a
x
z
x
)).,(),((
),(),(
txEtxxEa
atxxBatxB
LdB
yy
zz
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Waveb) Apply Ampere’s Law to a rectangle
in the zx-plane: Assume x is small enough that Ey ~ uniform
over surface.xa
t
txE
dt
d yE
),(
E
B
E
B
x x x
y
z
E, A
a
x
.),(),(),(
)),(),(( 0000 t
txE
x
txBxa
t
txEtxBtxxBa yzy
zz
x
z
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
4.4. Derivation of Solution: Plane WaveDerivation of Solution: Plane Wavec) Now take partial time and space derivatives
of both equations:
.),(),(
.),(),(
2
22
2
2
00
2
x
txE
tx
txB
t
txE
xt
txB
yz
yz
E
B
E
B
x x x
y
z z
.),(),(
2
2
002
2
t
txE
x
txE yy
This is the wave equation withv = ()-1/2 = c! (II.B.7)
(II.B.6)
II. EM Waves B.Maxwell’s II. EM Waves B.Maxwell’s EquationsEquations
5. Sinusoidal Wavesa) A more accurate representation of EM Wavesb) Plane waves can be a good approximationc) For wave propagating in the +x-direction:
d) E,B in phase, follow RHR: c, E, B
ktkxBB
jtkxEE
ˆ)cos(
ˆ)cos(
max
max
(II.B.8)
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
+++
t = 0: Charge placed on metal rods connected to anAC generator.
---
EV
II. EM WavesII. EM Waves
F. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
++
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
--
E
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
+
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
-
E
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
t = 0 to T/4: Rods neutralize, and E decreases to 0.Note: Initial E propagates away from arrayat speed c.
E = 0 at t = T/4.
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation-
t = T/4 to T/2: E reverses direction and grows.
+
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
--
t = T/4 to T/2: E reverses direction and grows.
++
II. EM WavesII. EM Waves
+++
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
---
t = T/4 to T/2: E reverses direction and grows.
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
a) Accelerating charges radiate energy as EM waves
b) Oscillating voltage => accelerates charge => EM radiation
---
+++
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave---
+++
v = c.
B
End result: A transverse wave of E propagatingat speed v = (00)-1/2 = c.
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
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B
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
+
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
II. EM WavesII. EM Waves
C. The Production of Electromagnetic Waves3. Antennae
c) Oscillating E => Oscillating B wave
Top View
-c
* E and B perpendicular to each other.* E and B perpendicular to v.* E and B in phase.
II. EM WavesII. EM Waves
II. EM WavesII. EM WavesD. Properties of EM Waves
1. Field strengths of EM waveE/B = c.
(II.D.1)2. Poynting Vector: Energy Flow Rate Vector
3. Power and Intensity: P= S per unit area, I = S(avg)
I = EmaxBmax/(20), (II.D.2)
= E2max/(20c) = B2
max(c/20).
BES
1
D. Properties of EM Waves
4. Radiation Pressurep = I/c (complete absorption) (II.D.3)p = 2I/c (complete reflection) (II.D.4)
5. EM waves in mattern = c/v = “index of refraction”
(II.D.5)
II. EM WavesII. EM Waves
E. The Electromagnetic Spectrum1. Units
a) Angstrom (Å) = 10-10 mb) Nanometer (nm) = 10-9 mc) Micron (m) = 10-6 m
2. Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays
3. VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet
(large wavelength to small)
II. EM WavesII. EM Waves
E. The Electromagnetic Spectrum1. Units
a) Angstrom (Å) = 10-10 mb) Nanometer (nm) = 10-9 mc) Micron (m) = 10-6 m
2. Radio, Microwave, Infrared, Visible, Ultraviolet, X-rays, Gamma rays
3. VISIBLE: “ROYGBIV” = Red, Orange, Yellow, Green, Blue, Indigo, and Violet
(large wavelength to small)
II. EM WavesII. EM Waves
A. Working Definitions1. Diffraction occurs when light source is not a
perfect point source and wave encounters a sharp edge.
2. Diffraction is essentially an example of interference between a large (continuous) distribution of sources.
3. Limits resolution of instruments—but also can be used to separate multi-chormatic light.
III. DiffractionIII. Diffraction
4. Spreading of wave from its initial line of travel
No diffraction
III. DiffractionIII. Diffraction
4. Spreading of wave from its initial line of travel
Diffraction
III. DiffractionIII. Diffraction
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
Unresolved pointof light
III. DiffractionIII. Diffraction
5. Diffraction occurs when light passes through a narrow opening, around obstacles and at sharp edges.
a) Application: Calculating stellar diameters by lunar occultation
III. DiffractionIII. Diffraction
Resolved diffractionpattern: spacing of fringes=> width of star
